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Properties

Label 2-1044-348.155-c0-0-1
Degree $2$
Conductor $1044$
Sign $-0.658 + 0.752i$
Analytic cond. $0.521023$
Root an. cond. $0.721819$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.532 − 0.846i)2-s + (−0.433 + 0.900i)4-s + (0.442 − 1.93i)5-s + (0.993 − 0.111i)8-s + (−1.87 + 0.656i)10-s + (0.678 − 0.541i)13-s + (−0.623 − 0.781i)16-s + (−0.314 + 0.314i)17-s + (1.55 + 1.23i)20-s + (−2.65 − 1.27i)25-s + (−0.819 − 0.286i)26-s + (0.993 + 0.111i)29-s + (−0.330 + 0.943i)32-s + (0.433 + 0.0990i)34-s + (0.211 + 1.87i)37-s + ⋯
L(s)  = 1  + (−0.532 − 0.846i)2-s + (−0.433 + 0.900i)4-s + (0.442 − 1.93i)5-s + (0.993 − 0.111i)8-s + (−1.87 + 0.656i)10-s + (0.678 − 0.541i)13-s + (−0.623 − 0.781i)16-s + (−0.314 + 0.314i)17-s + (1.55 + 1.23i)20-s + (−2.65 − 1.27i)25-s + (−0.819 − 0.286i)26-s + (0.993 + 0.111i)29-s + (−0.330 + 0.943i)32-s + (0.433 + 0.0990i)34-s + (0.211 + 1.87i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ -0.658 + 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8029813640\)
\(L(\frac12)\) \(\approx\) \(0.8029813640\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.532 + 0.846i)T \)
3 \( 1 \)
29 \( 1 + (-0.993 - 0.111i)T \)
good5 \( 1 + (-0.442 + 1.93i)T + (-0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.974 - 0.222i)T^{2} \)
13 \( 1 + (-0.678 + 0.541i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.314 - 0.314i)T - iT^{2} \)
19 \( 1 + (0.781 - 0.623i)T^{2} \)
23 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.433 + 0.900i)T^{2} \)
37 \( 1 + (-0.211 - 1.87i)T + (-0.974 + 0.222i)T^{2} \)
41 \( 1 + (1.27 + 1.27i)T + iT^{2} \)
43 \( 1 + (0.433 + 0.900i)T^{2} \)
47 \( 1 + (0.974 + 0.222i)T^{2} \)
53 \( 1 + (1.65 + 0.376i)T + (0.900 + 0.433i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-1.00 - 0.351i)T + (0.781 + 0.623i)T^{2} \)
67 \( 1 + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (1.43 + 0.900i)T + (0.433 + 0.900i)T^{2} \)
79 \( 1 + (-0.974 + 0.222i)T^{2} \)
83 \( 1 + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.831 - 1.32i)T + (-0.433 + 0.900i)T^{2} \)
97 \( 1 + (-0.623 + 0.218i)T + (0.781 - 0.623i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.876868460630035498565167101284, −8.814978137286584067503667478652, −8.595957006899364516377763448428, −7.81040503328880102157568227482, −6.36375172813579545279144301529, −5.19929905263569218016562583143, −4.55323981461203066325922466858, −3.47480480133209186555873373396, −1.96636185940918698484803254968, −0.964008431630797517141472047601, 1.92105256490263735489117875203, 3.13424363937103848041224460502, 4.37808495065731905579949914082, 5.75865257280227967556654054213, 6.38093587458197507349637250059, 6.99913109688749101102787863489, 7.71028043060574355958754801646, 8.761414175621960896117722630719, 9.655193965361949566901778359598, 10.27669292286059595081901930358

Graph of the $Z$-function along the critical line